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Mathematica Idioms for Array
Processing
Mark Reeve
XonTech Inc.
Download talk material:
This tutorial introduces useful Mathematica idioms and
utilities primarily for
numeric array processing. When treated in the serial, looping manner
of traditional
programming languages, Mathematica can appear slow and
cumbersome; however, elegant
and speedy performance can be recovered by using a holistic, parallel,
array-based
approach. Whereas some Mathematica texts discuss the
application of the basic
operators to one- or perhaps two-dimensional problems, very often a
problem will demand
three, four, five, or higher dimensions, leaving the would-be solver
floundering in
unfamiliar territory and perhaps ultimately resorting to awkward
looping constructs. Mathematica's
operator set is shown to be fully scalable to higher dimensions in a
straightforward way.
The goal is to catalyze the formation of a "matrix mindset"
in the
Mathematica practitioner, enabling the practitioner to decompose a
problem into its
natural fundamental constituents, and then to swiftly and confidently
dispose of each in a
few elegant, easily debugged lines of code.
The tutorial will first
lightly cover the
basics: Map, Apply, MapThread, generalized Transpose, etc.
It will then introduce some useful idioms by posing a few problems,
breaking each into
sections, and solving each section with an idiom. The specific
functionality addressed
will include: idioms for concatenating multidimensional arrays along
various dimensions,
reducing values from arrays along specific dimensions, robust and
efficient ways of
replacing nonconformable chunks of arrays, optimum numeric Boolean
operators, masking and
treating only elements satisfying a mask along the lines of Fortran's
WHERE utility, and
others as time permits. Wherever possible, a packed array
implementation will be
contrasted with the traditional approach.
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